# roulette wheel sequence

Is the sequence of numbers around a European roulette wheel (the integers from 0 to 36 inclusive) random or is there a pattern to it? It is said to have been devised by Pascal, which might be thought to argue against randomness. If one starts at zero and proceeds clockwise, the sequence is: 0-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18 -29-7-28-12-35-3-26

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## 1 Answer

This is a bit long for a comment:

Note that the simple chances (red-black, pair-impair, passe-manque) are designed to be almost independent (e.g. half of the evens are red and so on), though we cannot have full independence as $36$ is not a multiple of $8$. This is achieved among others by ensuring that exactly one of $n$ and $n+1$ is red and one is black (except for 10,11 both black, 18,19 both red, 28,29 both black). Now the arrangement of numbers on the wheel tries to make adjacent fields as different as possible (so even if you could predict the outcome up to one field, you don't win). Most strikingly, we have a red-black pattern. But the sequence also flips between passe and manque throughout. Only the odd-even sequence is not that nice. There might be better sequences if one aimed at also avoiding odd-odd and even-even as much as possible. But it seems that this sequence also somewhat balances the dozen and column chances. (Then again, maybe Pascal simply sorted the numbers into four urns representing colour/half combinations and drew them in sequence; one would need some original source to fill the details).

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